Newest Questions
159,026 questions
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Integer division: the length of the repetitive sequence after the decimal point
When dividing two integers, there may be an infinite sequence of digits after the decimal point (e.g. in the cases of 1/3, 1/7 etc).
As far as I know, if the two numbers divided are integers, this ...
- 159
22
votes
8
answers
2k
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Points and lines in the plane
Does a positive real number $k\geq1$ exist such that for every finite set $P$ of points in the plane (with the property that no three points of $P$ lie on a common line and $|P|\geq3$), one can choose ...
- 319
9
votes
3
answers
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Why is the Euler characteristic of powers of a line bundle a polynomial in the power?
Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor ...
- 31.2k
238
votes
46
answers
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Most interesting mathematics mistake?
Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
13
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4
answers
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Short Introduction to Planar Algebras
Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.
- 590
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votes
2
answers
1k
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Higher vanishing cycles
The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
- 10.8k
27
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7
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Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 52.7k
13
votes
0
answers
825
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Kato's log motives
What are they and what are their intended uses? Does anyone have notes/slides of this talk?
I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
- 10.8k
2
votes
4
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2k
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Closest grid square to a point in spherical coordinates
I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on ...
- 386
43
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1
answer
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What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
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3
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Conjugation in SU(2)
For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
- 1,129
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Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some ...
- 118k
2
votes
8
answers
3k
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The core question of topology
As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces.
To answer this question, one defines various properties of a space such as ...
- 159
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1
answer
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Do homotopy pullbacks commute with homotopy orbits (in spaces)?
Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{...
- 25.2k
6
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2
answers
468
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Algorithms for semistable reduction of families of curves
This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a ...
- 156k
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4
answers
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Arithmetic progressions without small primes
The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...
- 156k
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Is there a version of the valuative criteria for separateness/properness for varieties?
What I had in mind was something like the following:
X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.
Is there a good reason why ...
- 4,612
6
votes
2
answers
554
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Embedding abelian categories to have enough projectives
Is it true that the pro-objects of an abelian category form a category with enough projectives?
In general, given an abelian category A, is there a canonical way to embed it a bigger abelian ...
- 25.6k
-5
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3
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Gaussian curvature and mean curvature. [closed]
Define Gaussian curvature for a nonorientable surface. Can you define mean curvature for a nonorientable surface?
- 17
33
votes
8
answers
5k
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triangulated vs. dg/A-infinity
Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions".
I have a rough idea why this is true ("don't ...
- 12.8k
80
votes
7
answers
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Teaching statements for math jobs?
What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...
22
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5
answers
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Describing the universal covering map for the twice punctured complex plane
As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map.
In a sense, this shows that the logarithm has ...
- 5,530
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votes
4
answers
1k
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cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
- 4,265
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2
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Bad Categorical Quotients
Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it ...
- 2,806
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7
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Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 10.5k
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7
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2k
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Hochschild/cyclic homology of von Neumann algebras: useless?
Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...
- 5,425
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votes
2
answers
2k
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What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
- 7,237
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4
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3k
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Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.
Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...
- 118k
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2
answers
1k
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roots of analytic functions
Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
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95
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82k
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Undergraduate Level Math Books [closed]
What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate ...
6
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1
answer
578
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Does there exist a sequence of groups whose representation theory is described by plane partitions?
More precisely, does there exist a sequence $G_1 < G_2 < \cdots$ of finite groups such that the irreducible representations of $G_n$ are parameterized by the plane partitions of total size $n$?
- 118k
14
votes
4
answers
3k
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References for syntomic cohomology
Could anyone point to good readable references for learning about syntomic cohomology?
- 5,637
9
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10
answers
2k
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What do models where the CH is false look like?
Additionally, is there any intuitive way to visualize the cardinalities that result?
- 2,615
14
votes
2
answers
2k
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Co-induction understanding
Hi,
I am studying coinduction(not induction) as part of a class on static analysis. Rummaging around the internet, I am simply not finding a clear, concise description of:
How coinduction actually ...
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4
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When is a map given by a word surjective?
Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
- 20.2k
38
votes
6
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"Points" in algebraic geometry: Why shift from m-Spec to Spec?
Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started ...
- 2,967
11
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1
answer
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An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
25
votes
4
answers
2k
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algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
- 24.1k
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9
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What is the Tutte polynomial encoding?
Pretty much exactly what it says on the tin. Let G be a connected graph; then the Tutte polynomial T_G(x,y) carries a lot of information about G. However, it obviously doesn't encode everything about ...
- 12.6k
7
votes
2
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1k
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Is the Fourier transform of $\exp(-\|x\|)$ non-negative?
Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?
- 5,094
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6
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3k
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Formal consequences of Riemann-Roch (multiple answers welcome)
This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...
- 11.3k
8
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1
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Mirror symmetry for noncompact Calabi-Yau manifolds
In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...
- 287
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Reference for the `standard' Tate curve argument.
I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
- 10.5k
2
votes
1
answer
493
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Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
24
votes
3
answers
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Euler characteristic of a manifold and self-intersection
This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
- 5,530
41
votes
7
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5k
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Simplicial objects
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
- 21k
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2
answers
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Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
29
votes
5
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Handling arXiv feeds to avoid duplicates
I subscribe to feeds from the arXiv Front for a number of subject areas, using Google Reader. This is great, but there is one problem: when a new preprint is listed in several subject categories, it ...
40
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7
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What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?
I have a few elementary questions about cup-products.
Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an ...
- 25.6k
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8
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Are good introductory/pedagogical problems in algebraic geometry rare?
I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a ...